In △ABC, the median AD is 6 cm and CB is 12 cm, measure of angle CAB is _________.
- A90
∘ - B30
∘ - C60
∘ - D120
∘
Solution & Step-by-step Explanation
In △ABC, AD is the median to the side BC (or CB).
Therefore, D is the midpoint of BC.
BD=CD=
2
BC
=
2
12
=6 cm
We are given that the length of the median AD=6 cm.
Thus, AD=BD=CD=6 cm.
In △ABD, since AD=BD, it is an isosceles triangle. Let ∠BAD=∠ABD=x.
In △ACD, since AD=CD, it is an isosceles triangle. Let ∠CAD=∠ACD=y.
The sum of angles in △ABC is:
∠A+∠B+∠C=180
∘
(x+y)+x+y=180
∘
2(x+y)=180
∘
x+y=90
∘
Since ∠CAB=x+y, we have ∠CAB=90
∘
.
(Alternatively, by Apollonius' theorem or the property of a right-angled triangle, the circumradius from the right-angle vertex to the hypotenuse is equal to half the hypotenuse, which satisfies the condition here).
Therefore, D is the midpoint of BC.
BD=CD=
2
BC
=
2
12
=6 cm
We are given that the length of the median AD=6 cm.
Thus, AD=BD=CD=6 cm.
In △ABD, since AD=BD, it is an isosceles triangle. Let ∠BAD=∠ABD=x.
In △ACD, since AD=CD, it is an isosceles triangle. Let ∠CAD=∠ACD=y.
The sum of angles in △ABC is:
∠A+∠B+∠C=180
∘
(x+y)+x+y=180
∘
2(x+y)=180
∘
x+y=90
∘
Since ∠CAB=x+y, we have ∠CAB=90
∘
.
(Alternatively, by Apollonius' theorem or the property of a right-angled triangle, the circumradius from the right-angle vertex to the hypotenuse is equal to half the hypotenuse, which satisfies the condition here).